Banach Spaces from Barriers in High Dimensional Ellentuck Spaces

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A new hierarchy of Banach spaces Tₖ(d,θ), k any positive integer, is constructed using barriers in high dimensional Ellentuck spaces DobrinenJSL15 following the classical framework under which a Tsirelson type norm is defined from a barrier in the Ellentuck space Argyros/TodorcevicBK. The following structural properties of these spaces are proved. Each of these spaces contains arbitrarily large copies of ℓ∞ⁿ, with the bound constant for all n. For each fixed pair d and θ, the spaces Tₖ(d,θ), k≥ 1, are ℓₚ-saturated, forming natural extensions of the ℓₚ space, where p satisfies dθ=d¹/ᵖ. Moreover, they form a strict hierarchy over the ℓₚ space: For any j<k, the space Tⱼ(d,θ) embeds isometrically into Tₖ(d,θ) as a subspace which is non-isomorphic to Tₖ(d,θ).